Framed instanton homology and concordance, II

We continue our study of the integer-valued knot invariants ν ♯ (K) ^ (K) and r 0 (K) r₀ (K), which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on K K. We first establish a “conjugation” symmetry for the decomposition of cobordism maps constructed in our earlier work, and use this to prove, among many other things, that ν ♯ (K) ^ (K) is always either zero or odd. We then apply these technical results to study linear independence in the homology cobordism group, to define an instanton Floer analogue ϵ ♯ (K) ^ (K) of Hom’s ϵ -invariant in Heegaard Floer homology, and to the problem of characterizing a given 3-manifold as Dehn surgery on a knot in S 3 S³.